Postgraduate Science

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Real Analysis: Sequences & Convergence

Cauchy sequences, completeness of ℝ, and uniform vs pointwise convergence.

A sequence $(a_n) \subset \mathbb R$ converges to $L$ if $\forall \varepsilon>0\,\exists N:\, n\geq N \Rightarrow |a_n - L|<\varepsilon$.

$(a_n)$ is Cauchy if $\forall \varepsilon>0\,\exists N:\, m,n \geq N \Rightarrow |a_m - a_n|<\varepsilon$. $\mathbb R$ is complete: every Cauchy sequence in $\mathbb R$ converges. $\mathbb Q$ is not — e.g., $a_n = (1 + 1/n)^n$ is Cauchy in $\mathbb Q$ but its limit $e \notin \mathbb Q$.

Bolzano–Weierstrass: every bounded sequence in $\mathbb R^n$ has a convergent subsequence.

For sequences of functions $f_n \to f$:

  • Pointwise: $\forall x,\, f_n(x) \to f(x)$. Doesn't preserve continuity or interchange limits/integrals.
  • Uniform: $\sup_x |f_n(x) - f(x)| \to 0$. Preserves continuity; allows swapping with limits/integrals.

Quiz

1. $\mathbb R$ is complete means:
2. Bolzano–Weierstrass says every bounded sequence in $\mathbb R^n$ has:
3. Uniform convergence of $f_n \to f$ on $[a,b]$ guarantees:
4. $\mathbb Q$ is dense in $\mathbb R$ means:
5. A series $\sum a_n$ converges absolutely when: